Abstract
Let be the product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that
as
where
is the spectral radius of the matrix
and
is some constant. Consequently, the elements of
can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some delicate limit theories which are quite different from those of simple random walks. Limit theories of both the tail and critical tail sequences of continued fractions play important roles in our studies.
2020 Mathematics Subject Classifications:
Acknowledgments
The authors thank the referees for their helpful comments and suggestions that improved the original manuscript. Also the authors would like to extend special thanks to Prof. W. M. Hong for introducing to us the Lamperti problem and Prof. Y. Chow for some useful comments.
Disclosure statement
No potential conflict of interest was reported by the author(s).